Let \(\mathbf {\Psi }^{(\cdot )}\)  and \(\mathbf {\Upsilon }^{(\cdot )}\)  be the LLR realizations for the channel information and the extrinsic information to be passed to the detector, respectively.
Note that the DL-SPDA accepts both \(\mathbf {\Psi }^{(\cdot )}\)  and \(\mathbf {\Upsilon }^{(\cdot )}\)  as the inputs.
Each realization of the channel information \(\mathbf {\Psi }^{(\cdot )}\)  can be generated by transmitting a known data sequence \(\mathbf {x}\) .
On the other hand, to generate the extrinsic information \(\mathbf {\Upsilon }^{(\cdot )}\) , we adopt the idea from the previous work in [1]}.
We assume that the extrinsic information \(\mathbf {\Theta }^\rho \)  follows a consistent Gaussian probability density function, i.e.,
\(\mathbf {\Theta }_i^\rho \sim \mathcal {N}\big {(}(-1)^{x_i} \sigma _E^2/2,\sigma _E^2\big {)}\)  for \(i \in \lbrace 1,...N\rbrace \) , where the variance \(\sigma _E^2\)  can be computed by using the inverse \(J\) -function [2]}, given the mutual information \(\mathcal {M}(\mathbf {\Theta }^\rho , \mathbf {x})\)  between the extrinsic information \(\mathbf {\Theta }^\rho \)  and the transmitted sequence \(\mathbf {x}\) .
To maintain the generality of the NN, we assume that the mutual information \(\mathcal {M}(\mathbf {\Theta }^\rho , \mathbf {x}))\)  is uniformly ranged between \( [0, 1]\) .
In practice, it is sufficient to consider only a set of mutual information \(\Omega = \lbrace 0, 1/(|\Omega |-1), ..., 1\rbrace \) , where \(|\Omega |\)  is the cardinality of \(\Omega \) .
Thus, a set of variances can be then determined as \(\Xi = \lbrace J^{-1}(\partial ), \ \text{for} \ \partial \in \Omega \rbrace \) .
Based on the variance set \(\Xi \)  and the known transmitted sequence \(\mathbf {x}\)  in the training phase, we are able to randomly generate a realization of the LLR sequence \(\mathbf {\Upsilon }^{(\cdot )}_i \sim \mathcal {N}\big {(}(-1)^{x_i} \sigma _E^2/2,\sigma _E^2\big {)}\) , for \(i \in \lbrace 1,...,N\rbrace \) .
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